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Let $t$ be an integer such that $tgeq 2$. Let $K_{2,t}^{(3)}$ denote the triple system consisting of the $2t$ triples ${a,x_i,y_i}$, ${b,x_i,y_i}$ for $1 le i le t$, where the elements $a, b, x_1, x_2, ldots, x_t,$ $y_1, y_2, ldots, y_t$ are all distinct. Let $ex(n,K_{2,t}^{(3)})$ denote the maximum size of a triple system on $n$ elements that does not contain $K_{2,t}^{(3)}$. This function was studied by Mubayi and Verstraete, where the special case $t=2$ was a problem of ErdH{o}s that was studied by various authors. Mubayi and Verstraete proved that $ex(n,K_{2,t}^{(3)})<t^4binom{n}{2}$ and that for infinitely many $n$, $ex(n,K_{2,t}^{(3)})geq frac{2t-1}{3} binom{n}{2}$. These bounds together with a standard argument show that $g(t):=lim_{nto infty} ex(n,K_{2,t}^{(3)})/binom{n}{2}$ exists and that [frac{2t-1}{3}leq g(t)leq t^4.] Addressing the question of Mubayi and Verstraete on the growth rate of $g(t)$, we prove that as $t to infty$, [g(t) = Theta(t^{1+o(1)}).]
Given $r$-uniform hypergraphs $G$ and $H$ the Turan number $rm ex(G, H)$ is the maximum number of edges in an $H$-free subgraph of $G$. We study the typical value of $rm ex(G, H)$ when $G=G_{n,p}^{(r)}$, the ErdH{o}s-Renyi random $r$-uniform hypergra
Let $mathrm{rex}(n, F)$ denote the maximum number of edges in an $n$-vertex graph that is regular and does not contain $F$ as a subgraph. We give lower bounds on $mathrm{rex}(n, F)$, that are best possible up to a constant factor, when $F$ is one of
Classical questions in extremal graph theory concern the asymptotics of $operatorname{ex}(G, mathcal{H})$ where $mathcal{H}$ is a fixed family of graphs and $G=G_n$ is taken from a `standard increasing sequence of host graphs $(G_1, G_2, dots)$, most
The hypergraph duality problem DUAL is defined as follows: given two simple hypergraphs $mathcal{G}$ and $mathcal{H}$, decide whether $mathcal{H}$ consists precisely of all minimal transversals of $mathcal{G}$ (in which case we say that $mathcal{G}$
Let the bipartite Turan number $ex(m,n,H)$ of a graph $H$ be the maximum number of edges in an $H$-free bipartite graph with two parts of sizes $m$ and $n$, respectively. In this paper, we prove that $ex(m,n,C_{2t})=(t-1)n+m-t+1$ for any positive int